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Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



Scaling effects on the periodic homogenization of a reaction-diffusion-convection problem posed in homogeneous domains connected by a thin composite layer

Authors: Vishnu Raveendran, Emilio N. M. Cirillo, Ida de Bonis and Adrian Muntean
Journal: Quart. Appl. Math. 80 (2022), 157-200
MSC (2020): Primary 35B27; Secondary 35Q92
Published electronically: December 6, 2021
MathSciNet review: 4360553
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Abstract | References | Similar Articles | Additional Information


We study the question of periodic homogenization of a variably scaled reaction-diffusion problem with non-linear drift posed for a domain crossed by a flat composite thin layer. The structure of the non-linearity in the drift was obtained in earlier works as hydrodynamic limit of a totally asymmetric simple exclusion process (TASEP) for a population of interacting particles crossing a domain with obstacle.

Using energy-type estimates as well as concepts like thin-layer convergence and two-scale convergence, we derive the homogenized evolution equation and the corresponding effective model parameters for a regularized problem. Special attention is paid to the derivation of the effective transmission conditions across the separating limit interface in essentially two different situations: (i) finitely thin layer and (ii) infinitely thin layer.

This study should be seen as a preliminary step needed for the investigation of averaging fast non-linear drifts across material interfaces—a topic with direct applications in the design of thin composite materials meant to be impenetrable to high-velocity impacts.

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Additional Information

Vishnu Raveendran
Affiliation: Department of Mathematics and Computer Science, Karlstad University, Sweden
ORCID: 0000-0001-5168-0841

Emilio N. M. Cirillo
Affiliation: Dipartimento di Scienze di Base e Applicate per l’Ingegneria, Sapienza Universit‘a di Roma, Italy
MR Author ID: 606246
ORCID: 0000-0003-3673-2054

Ida de Bonis
Affiliation: Università telematica Niccolò Cusano, Rome, Italy
MR Author ID: 1072338

Adrian Muntean
Affiliation: Department of Mathematics and Computer Science, Karlstad University, Sweden
MR Author ID: 684703
ORCID: 0000-0002-1160-0007

Keywords: Reaction-convection-diffusion equation, homogenization, thin layer, dimension reduction, Galerkin method, two scale convergence, effective transmission condition
Received by editor(s): July 18, 2021
Received by editor(s) in revised form: October 26, 2021
Published electronically: December 6, 2021
Additional Notes: The work of the first and fourth authors was partly supported by the project “Homogenization and dimension reduction of thin heterogeneous layers”, grant nr. VR 2018-03648 of the Swedish Research Council
Article copyright: © Copyright 2021 Brown University